Fuglede Kadison Determinant Research Articles

The degree one Laguerre–Pólya class and the shuffle-word-embedding conjecture

Abstract We discuss the class of functions, which are well approximated on compacta by the geometric mean of the eigenvalues of a unital (completely) positive map into a matrix algebra or more generally a type $II_1$ factor, using the notion of a Fuglede–Kadison determinant. In two variables, the two classes are the same, but in three or more noncommuting variables, there are generally functions arising from type $II_1$ von Neumann algebras, due to the recently established failure of the Connes embedding conjecture. The question of whether or not approximability holds for scalar inputs is shown to be equivalent to a restricted form of the Connes embedding conjecture, the so-called shuffle-word-embedding conjecture.

Vanishing first cohomology and strong 1-boundedness for von Neumann algebras

We obtain a new proof of Shlyakhtenko's result which states that if G is a sofic, finitely presented group with vanishing first \ell^2 -Betti number, then L(G) is strongly 1-bounded. Our proof of this result adapts and simplifies Jung's technical arguments which showed strong 1-boundedness under certain conditions on the Fuglede–Kadison determinant of the matrix capturing the relations. Our proof also features a key idea due to Jung which involves an iterative estimate for the covering numbers of microstate spaces. We also use the works of Shlyakhtenko and Shalom to give a short proof that the von Neumann algebras of sofic groups with Property (T) are strongly 1 bounded, which is a special case of another result by the authors.

Fuglede-Kadison determinants over free groups and Lehmer’s constants

Lehmer’s famous problem asks whether the set of Mahler measures of polynomials with integer coefficients admits a gap at 1. In 2019, Lück extended this question to Fuglede-Kadison determinants of a general group, and he defined the Lehmer’s constants of the group to measure such a gap.

On some Fuglede-Kadison determinant inequalities of operator means

On some Fuglede-Kadison determinant inequalities of operator means

Logarithmic submajorizations inequalities for operators in a finite von Neumann algebra

The aim of this paper is to study the logarithmic submajorizations inequalities for operators in a finite von Neumann algebra. Firstly, some logarithmic submajorizations inequalities due to Garg and Aulja are extended to the case of operators in a finite von Neumann algebra. As an application, we get some new Fuglede-Kadison determinant inequalities of operators in that circumstance. Secondly, we improve and generalize to the setting of finite von Neumann algebras, a generalized Hölder type generalized singular numbers inequality.

Harnack type inequalities for operators in logarithmic submajorisation

The aim of this paper is to study the Harnack type logarithmic submajorisation and Fuglede-Kadison determinant inequalities for operators in a finite von Neumann algebra. In particular, the Harnack type determinant inequalities due to Lin-Zhang [15] and Yang-Zhang [27] are extended to the case of operators in a finite von Neumann algebra.

Lehmer’s problem for arbitrary groups

We consider the problem whether for a group [Formula: see text] there exists a constant [Formula: see text] such that for any [Formula: see text]-matrix [Formula: see text] over the integral group ring [Formula: see text] the Fuglede–Kadison determinant of the [Formula: see text]-equivariant bounded operator [Formula: see text] given by right multiplication with [Formula: see text] is either one or greater or equal to [Formula: see text]. If [Formula: see text] is the infinite cyclic group and we consider only [Formula: see text], this is precisely Lehmer’s problem.

An extension of the Beurling-Chen-Hadwin-Shen theorem for noncommutative Hardy spaces associated with finite von Neumann algebras

In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling’s theorem for a continuous unitarily invariant norm α on a tracial von Neumann algebra (M, τ) such that α is one dominating with respect to τ . The role of H∞ is played by a maximal subdiagonal algebra A. In the talk, we first will show that if α is a continuous normalized unitarily invariant norm on (M, τ), then there exists a faithful normal tracial state ρ on M and a constant c > 0 such that α is a c times one norm-dominating norm on (M,ρ). Moreover, ρ(x) = τ(xg), where x ∈M , g is positive in L1(Z, τ), where Z is the center of M . Here c and ρ are not unique. However, if there is a c and ρ so that the Fuglede-Kadison determinant of g is positive, then Beurling-Chen-Hadwin-Shen theorem holds for L(α)(M, τ). The key ingredients in the proof of our result include a factorization theorem and a density theorem for for L(α)(M,ρ). Talk time: 2016-07-19 03:00 PM— 2016-07-19 03:20 PM Talk location: Cupples I Room 207

Isometries on Positive Definite Operators with Unit Fuglede-Kadison Determinant

In this paper we explore the structure of certain surjective generalized isometries (which are transformations that leave any given member of a large class of generalized distance measures invariant) of the set of positive invertible elements in a finite von Neumann factor with unit Fuglede-Kadison determinant. We conclude that any such map originates from either an algebra $^*$-isomorphism or an algebra $^*$-antiisomorphism of the underlying operator algebra.

Jensen's inequality in finite subdiagonal algebras

Let $\mathscr{M}$ be a finite von Neumann algebra with a faithful normal tracial state $\tau$ and $\mathfrak{A}$ be a finite subdiagonal subalgebra of $\mathscr{M}$ with respect to a $\tau$-preserving faithful normal conditional expectation $\Phi$ on $\mathscr{M}$. Let $\Delta$ denote the Fuglede-Kadison determinant corresponding to $\tau$. For $X \in \mathscr{M}$, define $|X| := (X^*X)^{\frac{1}{2}}$. In 2005, Labuschagne proved the so-called Jensen's inequality for finite subdiagonal algebras i.e. $\Delta(\Phi(A)) \le \Delta(A)$ for an operator $A \in \mathfrak{A}$, thus resolving a long-standing open problem posed by Arveson in 1967. In this article, we prove the following more general result: $\tau(f( |\Phi(A)|)) \le \tau(f( |A|))$ for $A \in \mathfrak{A}$ and any increasing continuous function $f : [0, \infty) \to \mathbb{R}$ such that $f \circ \exp$ is convex on $\mathbb{R}$. Under the additional hypotheses that $A$ is invertible in $\mathscr{M}$ and $f \circ \exp$ is strictly convex, we have $\tau(f( |\Phi(A)|)) = \tau(f( |A|))$ $\Longleftrightarrow \Phi(A) = A$. As an application, we show that for $A \in \mathfrak{A}$ the point spectrum of $A$ is contained in the point spectrum of $\Phi(A)$, though such a conclusion does not hold in general for their spectra.

On a class of determinant preserving maps for finite von Neumann algebras

Let R be a finite von Neumann algebra with a faithful tracial state τ and let Δ denote the associated Fuglede–Kadison determinant. In this paper, we characterize all unital bijective maps ϕ on the set of invertible positive elements in R which satisfyΔ(ϕ(A)+ϕ(B))=Δ(A+B). We show that any such map originates from a τ-preserving Jordan ⁎-automorphism of R (either ⁎-automorphism or ⁎-anti-automorphism in the more restrictive case of finite factors). In establishing the aforementioned result, we make crucial use of the solutions to the equation Δ(A+B)=Δ(A)+Δ(B) in the set of invertible positive operators in R. To this end, we give a new proof of the inequalityΔ(A+B)≥Δ(A)+Δ(B), using a generalized version of the Hadamard determinant inequality and conclude that equality holds for invertible B if and only if A is a nonnegative scalar multiple of B.

Joint spectrum and the infinite dihedral group

For a tuple A = (A 1,A 2, ... ,A n ) of elements in a unital Banach algebra B, its projective joint spectrum P(A) is the collection of z ∈ ℂ n such that the multiparameter pencil A(z) = z 1 A 1 + z 2 A 2 + ... + z n A n is not invertible. If B is the group C*-algebra for a discrete group G generated by A 1,A 2, ... ,A n with respect to a representation ρ, then P(A) is an invariant of (weak) equivalence for ρ. This paper computes the joint spectrum of R = (1, a, t) for the infinite dihedral group D ∞ = 〈a, t | a 2 = t 2 = 1〉 with respect to the left regular representation λ D∞, and gives an in-depth analysis on its properties. A formula for the Fuglede–Kadison determinant of the pencil R(z) = z 0 + z 1 a + z 2 t is obtained, and it is used to compute the first singular homology group of the joint resolvent set P c (R). The joint spectrum gives new insight into some earlier studies on groups of intermediate growth, through which the corresponding joint spectrum of (1, a, t) with respect to the Koopman representation ρ (constructed through a self-similar action of D ∞ on a binary tree) can be computed. It turns out that the joint spectra with respect to the two representations coincide. Interestingly, this fact leads to a self-similar realization of the group C*-algebra C*(D ∞). This self-similarity of C*(D ∞) manifests itself in some dynamical properties of the joint spectrum.

Fuglede–Kadison Determinants and Sofic Entropy

We relate Fuglede-Kadison determinants to entropy of algebraic actions of sofic groups in essentially complete generality. This generalizes recent results of Hanfeng Li and Andreas Thom from the amenable case to the sofic case, as well as results of David Kerr, Hanfeng Li, and Lewis Bowen in the residually finite case. Moreover, the proof given is the first calculation of entropy of algebraic actions to avoid a nontrivial determinant approximation. We apply our results to a problem of Christopher Deninger, generalizing results of Hanfeng Li and David Kerr. Finally, we show that in many cases, finiteness of topological entropy for algebraic actions of sofic groups is equivalent to having metric mean dimension zero.

Fuglede–Kadison determinant: theme and variations

We review the definition of determinants for finite von Neumann algebras, due to Fuglede and Kadison [Fuglede B, Kadison R (1952) Ann Math 55:520-530], and a generalization for appropriate groups of invertible elements in Banach algebras, from a paper by Skandalis and the author (1984). After some discussion of K-theory and Whitehead torsion, we indicate the relevance of these determinants to the study of -torsion in topology. Contents are as follows:1. The classical setting. 2. On von Neumann algebras and traces. 3. Fuglede-Kadison determinant for finite von Neumann algebras. 4. Motivating question. 5. Brief reminder of K0, K1, K1(top), and Bott periodicity. 6. Revisiting the Fuglede-Kadison and other determinants. 7. On Whitehead torsion. 8. A few lines on L2-torsion.

On the spectral density function of the Laplacian of a graph

Let X be a finite graph. Let |V| be the number of its vertices and d be its degree. Denote by F1(X) its first spectral density function which counts the number of eigenvalues ≤λ2 of the associated Laplace operator. We provide an elementary proof for the estimate F1(X)(λ)−F1(X)(0)≤2⋅(|V|−1)⋅d⋅λ for 0≤λ<1 which has already been proved by Friedman (1996) [3] before. We explain how this gives evidence for conjectures about approximating Fuglede–Kadison determinants and L2-torsion.

Fuglede-Kadison determinants for operators in the von Neumann algebra of an equivalence relation

We calculate the Fuglede-Kadison determinant for operators of the form ∑n i=1 MfiLgi where Lgi are unitaries or partial isometries coming from Borel (partial) isomorphisms gi on a probability space which generate an ergodic equivalence relation, and Mfi are multiplication operators. We obtain formulas for the cases when the relation is treeable or the fi’s and gi’s satisfy some restrictions.

Entropy, determinants, and $L^2$-torsion

We show that for any amenable group \Gamma and any Z\Gamma-module M of type FL with vanishing Euler characteristic, the entropy of the natural \Gamma-action on the Pontryagin dual of M is equal to the L2-torsion of M. As a particular case, the entropy of the principal algebraic action associated with the module Z\Gamma/Z\Gamma f is equal to the logarithm of the Fuglede-Kadison determinant of f whenever f is a non-zero-divisor in Z\Gamma. This confirms a conjecture of Deninger. As a key step in the proof we provide a general Szeg\H{o}-type approximation theorem for the Fuglede-Kadison determinant on the group von Neumann algebra of an amenable group. As a consequence of the equality between L2-torsion and entropy, we show that the L2-torsion of a non-trivial amenable group with finite classifying space vanishes. This was conjectured by L\"uck. Finally, we establish a Milnor-Turaev formula for the L2-torsion of a finite \Delta-acyclic chain complex.

Compact group automorphisms, addition formulas and Fuglede-Kadison determinants

For a countable amenable group \Gamma and an element f in the integral group ring Z\Gamma being invertible in the group von Neumann algebra of \Gamma, we show that the entropy of the shift action of \Gamma on the Pontryagin dual of the quotient of Z\Gamma by its left ideal generated by f is the logarithm of the Fuglede-Kadison determinant of f. For the proof, we establish an \ell^p-version of Rufus Bowen's definition of topological entropy, addition formulas for group extensions of countable amenable group actions, and an approximation formula for the Fuglede-Kadison determinant of f in terms of the determinants of perturbations of the compressions of f.

Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents

For an ergodic automorphism γ of a probability space (Ω,A, P ) consider the semidirect product von Neumann algebra R = L(Ω) ⋊γ Z. Let U be the unitary operator corresponding to the action of γ. Then the finite sums of the form Φ = ∑ i aiU i with ai in L (Ω) form a strongly dense subalgebra of R. In this paper we calculate the Fuglede–Kadison determinant detR Φ for a certain class of elements Φ. The answer is given in terms of the Ljapunov exponents χi of a matrix AΦ over the ring L (Ω) which is attached to Φ by a simple receipe. The formula has the form

Identities and Inequalities for Tree Entropy

The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses Fuglede–Kadison determinants, while another uses effective resistance. We use the latter to prove that tree entropy respects stochastic domination. We also prove that tree entropy is non-negative in the unweighted case, a special case of which establishes Lück's Determinant Conjecture for Cayley-graph Laplacians. We use techniques from the theory of operators affiliated to von Neumann algebras.